None of the challenges immediately derailed the recounts in those states, but they promised to complicate them with more legal wrangling by Mr. Trump, groups supportive of him, state officials and Jill Stein, the Green Party presidential candidate. Ms. Stein initiated the recounts and a successful fund-raising drive after suggesting that voting machines were susceptible to hacking.

On Friday, Mr. Trump filed a lawsuit in the Michigan Court of Appeals in an attempt to block the recount there, which had not yet begun. “If the Bureau of Elections moves forward with the recount, it will waste the State’s scarce resources, create a logistical nightmare for counties across the State, and assure that Michigan’s Electoral College voters will not be counted,” the filing said.

Bill Schuette, Michigan’s attorney general, filed a separate lawsuit in a bid to halt the recount, saying that it put the state’s voters at risk of “paying millions and potentially losing their voice in the Electoral College in the process.”

In his court filing, Mr. Schuette, a Republican who is widely mentioned as a possible candidate for governor in 2018, said, “This court cannot allow a dilatory and frivolous request for a recount by an aggrieved party to silence all Michigan votes for president.’’

In Wisconsin, a lawsuit against the state Elections Commission was filed Thursday in Federal District Court by the Great America PAC, the Stop Hillary PAC and Ronald R. Johnson, a Wisconsin resident. The lawsuit said that the recount could “unjustifiably cast doubt” on Mr. Trump’s victory in that state.

The plaintiffs argued that the recount, which began across the state’s 72 counties on Thursday morning, should be halted immediately, in part because there was a substantial chance that it cannot be accurately completed by mid-December. Results must be certified before the Electoral College meets on Dec. 19. In 2011, a statewide recount took close to a month.

A federal judge said Friday that he would not halt the recount, but allowed the lawsuit to proceed.

Lawyers for Mr. Trump and his allies are also seeking to halt legal proceedings by Ms. Stein to contest the statewide election results in Pennsylvania.

Hillary Clinton’s campaign has participated only lightly, paying for lawyers to be present at recount sites. Still, the recounts have generated excitement among some of her supporters, hoping that the small margins in the three traditionally Democratic states might swing to Mrs. Clinton in a recount.

But Edward B. Foley, director of the Election Law Project at Ohio State University’s Moritz College of Law, said there was no comparison between this recount and the 2000 Florida impasse between George W. Bush and Al Gore. Then, he said, “it was quite plausible that Gore might prevail in a recount.’’

The chance that these state recounts could reverse the outcome of the 2016 election, Mr. Foley said, was “essentially zero or infinitesimal.”

Ms. Stein, in a statement, said the challenges to the recounts were an effort to put “party politics above country.”

“Trump’s desperate attempts to silence voter demands for recounts raise a simple question: Why is Donald Trump afraid of these recounts?” she said.

In Michigan, where the recount is still pending, the Board of State Canvassers on Friday heard a formal objection from Mr. Trump and his campaign to plans to recount that state’s 4.8 million votes.

Lawyers for Mr. Trump told the state board, which is made up of two Democrats and two Republicans, that it should not allow a new count, describing it as needless, too expensive and not required under Michigan law. Ms. Stein, who got 51,463 votes in Michigan, hardly met the state’s standard as an “aggrieved” party, the lawyers argued.

But a lawyer for Ms. Stein, Mark Brewer, said Mr. Trump’s campaign was making a “desperate attempt” to avoid a recount. Of the three states where recounts are contemplated, Michigan had the smallest gap between Mr. Trump and Mrs. Clinton: 10,704 votes.

“I would remind everybody that the original source of the allegation that this election was rigged was Mr. Trump,” Mr. Brewer, a former chairman of Michigan’s Democratic Party, told the board.

In the end, after several tense exchanges over the practical value and cost of a recount, the Board of State Canvassers split, 2 to 2, along party lines, meaning the recount objection failed.

Barring a court order in Mr. Schuette’s case or some other legal development, Michigan, which has not seen a statewide recount in half a century, could begin a hand recount by Wednesday.

In Pennsylvania, Ms. Stein’s recount campaign is operating along dual tracks, with its challenge of the statewide results proceeding along with petitions to recount specific voting districts.

Lawrence J. Tabas, general counsel of the Republican Party of Pennsylvania, said in an interview on Friday that Ms. Stein’s lawyers had fallen short of demonstrating that there was fraud or illegal action in the Nov. 8 election.

“They know they have no claim,” he said. “This action by Jill Stein and her supporters — I couldn’t even call it a Hail Mary pass, because that would be insulting to the Hail Mary pass,” said Mr. Tabas, who along with other lawyers, submitted a lengthy court brief on Thursday.

On Friday afternoon in Philadelphia, teams of observers were monitoring election workers as they rechecked counts on electronic voting machines used in 75 precincts, a small fraction of the nearly 1,700 across the city.

The groups — which included representatives from each of the parties and candidates — quietly made their way from machine to machine in a dimly lit warehouse in North Philadelphia where the city stores its 4,000 machines. Philadelphia has been using the machines, which record votes on cartridges, since the early 2000s.

The review mirrored the steps that automatically take places after elections, said Al Schmidt, a city commissioner and a Republican, who said he did not expect the vote tally to change.

“We are doing the exact same thing that we did during the computation process,” he said. “We are just doing it once again.”

Rich Garella, 50, a Stein supporter who petitioned for the recount and was serving as a monitor on Friday, said the effort felt worthwhile, but he had hoped that election officials would approve a more rigorous forensic audit of the machines, which Ms. Stein had requested.

“We as voters don’t really understand how these machines work,” Mr. Garella said. “We don’t understand how they are programmed.”

He predicted that one result of the recount effort, at least in Philadelphia, could be to shed light on the unseen part of the voting system.

“What we may end up showing from this is just that we are not being allowed to see how our election works and to see that our votes are being counted properly,” he said.